| Metamorphosis of the Cube (1999) | |||||||||||||||
Abstract | |||||||||||||||
| Introduction The foldings and unfoldings shown in this video illustrate two problems: (1) cut open and unfold a convex polyhedron to a simple planar polygon; and (2) fold and glue a simple planar polygon into a convex polyhedron. A convex polyhedron can always be unfolded, at least when we allow cuts across the faces of the polyhedron. One such unfolding is the star unfolding [2]. When cuts are limited to the edges of the polyhedron, it is not known whether every convex polyhedron has an unfolding [3], although there is software that has never failed to produce an unfolding, e.g., HyperGami [6]. The problem of folding a polygon to form a convex polyhedron is partly answered by a powerful theorem of Aleksandrov. The following section describes this in more detail. After that, we explain the examples that are shown in the video, and discuss some open problems. 2 Aleksandrov's conditions Aleksandrov's theorem [1] concerns the realization of a "po | |||||||||||||||
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